I'm not going to try to address the scenarios, at least not yet. Instead, I'm going to try looking at things from a tech basis, seeing how each of the two sides match up to the Battletech universe's Mechs and Mechwarriors.
Let's start with the most important limitation of Battletech: the FTL barrier and JumpShip technology. Neither the UE nor the VE have any FTL technology; the VE once had access to the OP-tech known as the Hyper Gate, a form of FTL that presumably kicks JumpDrives in the balls, but it was destroyed and there's no evidence that they have the ability to recreate that tech. This means that in any world-hopping situation, both the UE and the VE could take control of their local space, but have no ability to project force beyond their local sphere. This means that until they can obtain their own JumpDrives, they'd both be forced to maintain a defensive-only posture. Both of them are at a tech level where they could reverse-engineer a JumpDrive and build their own JumpShips (VE could probably do it faster and better), but it'll take time for both of them.
Other than that, Battletech's unique combat technologies are 1) myomer; 2) neurohelmets; and 3) fusion plants. (HPGs don't count for this analysis, although I agree with the thrust of Loki's point.) All of these technologies are things that both sides could almost certainly reverse-engineer, although whether they'd want to is a different story. I'll come back to this later.
Moving on, we've got the United Earth forces. The UE's Kataphracts (UEKs) are... actually a pretty good match-up for Battletech mechs. Ryven's post covers most of the salient points; modular loadouts, light armor, good maneuverability. Size-wise, they're around the Heavy weight class at 13.5 meters. Their biggest edge over Battlemechs is their ability to operate in space, although there's substantially less call for them to do so in BT. Their biggest weakness is their comparatively low armor and weapons counts.
Myomers might be useful for the UEK due to their load-bearing abilities, but would involve massive retooling and redesigns, so unless they could directly jump to Triple Strength Myomer somehow, they would probably stick to mechanical motivators. Neurohelmets, on the other hand, would be much easier to implement, especially since the AI-Equipped Cyborg Eyeball is significantly more advanced. No idea how much advantage they could derive from neurohelmets, though; UEKs already have auto-balancing systems, but since the UE could probably figure out how to make Star League-grade neurohelmets or better, they might be able to bestow CyberEye capabilities on their pilots without brain surgery. Who knows.
Could UEKs be upgraded to use fusion drives? Probably. If I remember correctly, their power plants are mounted on backpacks that can be ejected in an emergency, so it would just be a question of switching backpacks. Whether they would want to do so is a different question; fission plants require exotic elements for operation that fusion plants don't, but there doesn't seem to be a huge difference in field performance. Battlemechs can operate in the field for far longer than Next, there are myomers. Neither the UE or the VE use anything like this, and it doesn't seem to have set them back any. Both their tech bases rely on mechanical movement, not contractive. Myomer's big advantage is its load-bearing ability, letting Battlemechs carry their heavy armor and weapons. This gives them a major advantage over UE thanks to this, so fusion drives would be superior for garrison duties, but I suspect that current UEK drives might actually be better for front-line deployments thanks to their lower heat output.
The big question is how efficient the UE is at manufacturing UEKs, whether they're Sleipnir or Areion models. BT's limitation is the ability to replenish mechs at a sufficient level of quality; they're limited by LosTech, whereas the UE doesn't have that problem. If the UE can crank out enough UEKs fast enough using current designs, they could probably maintain parity with most invading forces.
On the other side of the coin, we have the Versian Empire and its Super Kataphracts (let's call them VESKs). This really is the only way to describe them; like Ryven said, these things are capable of BS-level OP-tech. Their Landing Castles are stupidly big; at 2 kilometers in length, they make BT's 131-meter Overlord dropships look dinky in both carrying capacity and armaments. Their VESKs can pull off such impossible effects as infinite self-duplication, dimensional invincibility, gravitic manipulation, entropic manipulation, "all the lasers", and many more. In any battle between any VESK and a full battalion of Battlemechs, the VESK would almost certainly be the victor. Broadly speaking, the VE points and laughs at BT capabilities.
In terms of useful BT tech, the only thing the VE would care about is JumpDrive technology. Fusion is laughable next to the power of Aldnoah, and myomers might make an amusing curiosity. Neurohelmet tech might be interesting, but most VESKs wouldn't need it and those that could use it already have significantly superior methods of mech control.
Unfortunately, the VE has several limitations. First, there's the numbers problem. The VE doesn't have mass-production units; the closest thing they have is the Stygis, and those only have speed to protect them. The loss of a single VESK is a huge blow to the VE, and you better believe that if one brilliant teenager could spot the flaws in the various VESK designs, then so could the more experienced Mechwarriors in BT. Which brings us to the next big problem: the VESK designs are... just really bad. Yeah, I said it. With a very few exceptions, the VESKs are entirely dependent on their one superpower. Find a way around that, and they have nothing else with which to defend themselves. I mean, the infinite self-duplication VESK's only weapons are two knuckledusters. It punches. That's its only attack. And that's saying nothing about the actual physical designs, which are impractical at best.
If we step back to the other side of the fourth wall, I can say that the numerous and massive design flaws in the VESKs were there due to a combination of rule-of-cool and the necessity of making the VESKs beatable by AZ's hero. The only in-universe explanations are that either the different Aldnoah effects somehow interfere with conventional weapons and armor, or the VE engineers had drunk too much of the "Versian Superiority" Kool-Aid. Either way, the VESKs make for amazing shock-and-awe platforms, but would be slowly ground down in an actual campaign - as happened in AZ. And getting them replaced? Well, that's the third issue: Aldnoah Drives are a non-renewable resource. As far as we know, the VE has no ability to produce new Aldnoah Drives, only the technology that uses and channels their effects. ADs are the ultimate LosTech. On top of that, the number of AD users is kept artificially low by the Versian Royal Family. Losses could be replenished, but even more slowly than BT mechs, to say nothing of the already-small VE population as a limiting factor. All this means that without a serious build-up, the VE couldn't project force to more than a couple of nearby star systems even if they had FTL tech.
Overall, my feeling is that both the UE and VE could individually hold whatever system they appeared in (with the exception of Terra), but would be unable to expand their sphere of control beyond that range for a very long time. Any influence on the galactic stage would have to be obtained through diplomacy, which would require entirely different approaches and outcomes depending on which Successor State either of them appeared in.
And if they both appeared in Terra? Well, that would depend on what point in the BT timeline they appeared at, but honestly, my guess is that it would kick off a whole new Succession War.
